An Anti-Ramsey Theorem of k-Restricted Edge-Cuts

Abstract

Given a graph G (with multiple edges allowed), a k-restricted edge-cut of G is an edge-cut $R\subseteq E(G)$ of G such that every connected component in $G-R$ has order at least k. A graph G is $\lambda _k$-connected if k-restricted edge cuts exist. Given an edge-coloring of G, an edge-cut of G will be called heterochromatic if all its edges receive different colors. Given a graph G and an integer $k\ge 2$, let h(G, k) be the minimum integer p such that every p-coloring of the edges of G produces an heterochromatic k-restricted edge-cut of G. In this paper we prove that for a $\lambda _k$-connected graph G of size m, order $n\ge 3k-2$, and with no k-restricted edge-cut of size 1, we have that $m-n+ \min \{ \kappa _0(G)-2, t(G,k)\} +2 \le h(G,k) \le m-n+ t(G,k) +2$, where $\kappa _0(G)$ is the vertex-connectivity of G and t(G, k) is the maximum cardinality of a set $S\subseteq V(G)$ such that the connected components in G[S] have order at most $k-1$. As a corollary we see that $h(G,k) \le m-n+ \alpha (G)(k-1) +2$, with $\alpha (G)$ the independence number of G.